Pythonで学ぶ統計学の基礎

辻 真吾(@tsjshg)

みんなのPython勉強会#9

2016/2/8

自己紹介

  • 1975年8月3日生まれ(今年は大厄)
  • Pythonを使ったデータ解析が結構得意
  • 東京大学先端科学技術研究センターゲノムサイエンス分野
  • 1/15に切れた任期が3/31まで伸びました!
  • UdemyでPythonを使ったデータ解析のコースを翻訳しました(近日公開予定)

最近、注目を集める統計学

ビジネス書大賞2014年

大学入学以来20数年、統計学が分からない・・・

10年くらい前、1冊の本に出会う

松原先生と言えば、大学1年の時に統計の講義をとっていた!

なんと1ページ目に次のような1文が

実は、現在でもいまだに「統計学はよくわからない」という印象が残ったままである。

統計学の専門家にわからないものが、俺にわかる訳がない!

In [1]:
# 必要なものをimport
%matplotlib inline
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn

def set_fig():
    # プレゼン用
    plt.figure(figsize=(10,6))
    plt.xticks(fontsize=20)
    plt.yticks(fontsize=20)
In [25]:
set_fig()
# まずは乱数を発生させるところから
# 0から1の間の小数を1,000個(一様分布)
uniform_rand = np.random.uniform(0,1,1000) 
_ret = plt.hist(uniform_rand, bins = 50)
In [26]:
# 10個ずつまとめる(100行10列のデータに変形)
uniform_reshape = uniform_rand.reshape(100,10)

print(uniform_reshape.shape)
print(uniform_reshape)
(100, 10)
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    7.14146926e-01   4.39561652e-01   6.57552014e-01   4.56524711e-01
    8.61421016e-01   5.80489434e-02]
 [  4.06560970e-01   3.93827621e-01   5.60249001e-01   5.08952584e-01
    1.13203131e-01   2.94128243e-01   9.47348924e-01   4.25292566e-01
    6.37149031e-03   8.84449878e-01]
 [  3.60195151e-01   2.91276042e-02   7.72618172e-01   5.60991459e-01
    7.05997443e-01   1.24626355e-01   8.38762223e-01   3.87586377e-01
    8.40330687e-01   5.14717470e-01]
 [  8.86318636e-01   2.03720082e-01   9.67087093e-01   9.06472080e-02
    9.90987438e-02   3.31484253e-01   4.31239443e-01   1.49220822e-01
    4.10981387e-02   1.23194929e-01]
 [  2.40126113e-01   5.36107858e-01   1.48049848e-01   5.81401634e-01
    3.84888488e-01   9.64189582e-01   8.14638949e-01   4.46027874e-01
    3.11619778e-01   7.43892587e-01]
 [  7.15183060e-01   4.18650674e-01   9.16900123e-01   2.55653926e-01
    7.05398442e-02   7.61730643e-01   3.03179456e-01   3.16286333e-01
    2.75762957e-02   7.29636296e-01]
 [  7.10126513e-01   7.22104910e-01   9.01342866e-01   5.00197924e-01
    4.33152120e-02   9.86357430e-01   6.00353488e-01   2.71956159e-01
    7.21063806e-01   1.39846867e-01]
 [  9.86421713e-01   8.10400544e-01   1.01493816e-01   3.78392164e-01
    4.07687626e-01   9.38426395e-01   2.34503523e-01   6.84754703e-01
    5.46860234e-01   1.13924081e-01]
 [  5.76616207e-02   1.69214445e-01   9.40497761e-01   7.13344933e-01
    5.97386596e-01   1.77325825e-01   3.10864512e-01   7.19681654e-01
    1.84973142e-01   3.09682031e-01]
 [  6.53146715e-01   1.65844325e-02   4.49616642e-01   4.63864061e-01
    5.28687676e-01   3.92610311e-01   6.54854229e-01   5.62878841e-01
    3.35582572e-02   1.14637325e-01]
 [  5.13527846e-02   4.79516923e-01   6.86837383e-01   5.84762097e-01
    8.13292444e-02   4.04887604e-01   5.03866243e-01   5.48737942e-01
    3.36665353e-01   8.50705573e-01]
 [  5.06391888e-01   1.99807205e-01   5.56457307e-01   5.09297257e-01
    9.44579811e-01   1.74087330e-01   6.21774984e-01   6.65350182e-01
    5.51920730e-01   8.27762045e-01]
 [  1.68550000e-01   2.27610835e-01   1.17395788e-01   1.65576942e-01
    4.41184805e-01   2.45982248e-01   1.42759338e-01   9.08397514e-01
    4.98816660e-01   5.40776676e-01]
 [  6.35369683e-01   6.41260310e-01   7.99940149e-01   4.06273783e-01
    9.05012102e-01   3.81647876e-01   7.21158243e-01   2.95952354e-01
    8.61558490e-02   9.82895099e-02]
 [  2.55604582e-01   3.11873471e-01   5.53437095e-02   7.74498791e-01
    2.19105469e-01   7.66073548e-01   4.08322369e-01   1.54938338e-01
    6.29352549e-01   2.13994756e-01]
 [  9.85778502e-01   4.82004862e-01   3.83494142e-01   2.79825542e-01
    4.34622457e-01   4.26922782e-01   1.34971956e-01   2.82356198e-01
    3.88165017e-01   9.71409013e-01]
 [  4.22378952e-02   7.60801817e-01   4.84468598e-01   4.59863241e-01
    3.62759482e-01   4.49827386e-01   7.24454962e-01   7.25404778e-01
    7.90620238e-01   9.99087340e-02]
 [  4.06301111e-01   8.93455095e-01   2.38913482e-01   8.60351710e-01
    2.83478322e-01   4.42536585e-01   9.03051982e-01   4.06785369e-01
    6.51046918e-01   8.01189864e-01]
 [  8.26932898e-01   9.62355562e-01   4.78462892e-01   8.85657994e-03
    8.81350546e-01   4.17886007e-01   1.27199663e-01   7.50782199e-01
    9.80979081e-01   9.47872405e-01]
 [  6.75428099e-02   5.23128365e-01   4.41213684e-01   1.99579299e-01
    6.86487949e-01   3.82714462e-03   4.89676260e-01   6.39592544e-01
    6.76338399e-01   9.06912868e-02]
 [  8.92266045e-02   6.11850275e-01   6.56301718e-01   9.74636928e-01
    3.70095239e-01   4.43241237e-01   3.75618797e-01   4.57658998e-02
    3.57148195e-01   8.79082257e-01]
 [  9.40893273e-01   2.34827528e-01   1.46969551e-01   8.43478197e-01
    6.29800258e-01   6.46713017e-02   4.85463548e-01   4.21998497e-01
    1.84065996e-01   4.03600052e-02]
 [  3.92770514e-01   3.59242861e-02   7.80424191e-01   6.40771108e-01
    5.79175668e-01   4.16974914e-01   6.76258079e-01   2.28250366e-01
    5.67731166e-01   3.37784927e-01]
 [  7.79895384e-01   7.23893796e-02   6.85105138e-01   3.66882141e-01
    6.46803783e-01   4.21481369e-01   3.93782108e-01   4.50769798e-01
    6.14713482e-01   3.10731093e-01]
 [  8.78083469e-01   1.86853571e-01   3.61303402e-01   9.46667917e-01
    3.62527596e-01   7.28523661e-01   7.51055133e-01   2.58128652e-01
    1.98411130e-01   3.63565025e-01]
 [  7.46631905e-01   1.81006924e-01   7.35164878e-01   5.68448970e-01
    1.58735192e-01   5.72265734e-01   8.37714282e-01   3.08716039e-01
    1.42985408e-01   3.19057212e-01]
 [  9.17419082e-01   3.39851621e-01   1.71538254e-01   4.42358024e-01
    8.48472735e-01   5.99241055e-01   3.80710914e-01   9.27347621e-01
    2.31326207e-01   9.36315959e-01]
 [  3.61397511e-01   7.42627850e-01   5.44911127e-03   8.35850764e-01
    6.90746439e-01   8.81484573e-01   9.45888642e-01   9.92539213e-01
    1.64338459e-01   8.91302042e-01]]
In [27]:
# そして、その10個の平均をとる(100個のデータになる)
uniform_avg10 = uniform_reshape.mean(1)

print(uniform_avg10.shape)
print(uniform_avg10)
(100,)
[ 0.47368519  0.47583721  0.52277867  0.52239101  0.47174741  0.52062103
  0.38191706  0.47778701  0.33649237  0.34532942  0.61340561  0.58704056
  0.67284833  0.56624629  0.43570719  0.46743566  0.58937005  0.4538734
  0.39299709  0.65459217  0.49526204  0.4568108   0.66267305  0.56844084
  0.54858283  0.4775614   0.54180814  0.54668602  0.59504325  0.32411337
  0.52647702  0.63744935  0.46622217  0.43622435  0.42123508  0.46899132
  0.37680244  0.54295178  0.68477773  0.61006972  0.43570479  0.59144038
  0.38879381  0.61461549  0.54388166  0.46817779  0.50773439  0.41155671
  0.47326181  0.47915055  0.55110739  0.55387816  0.44951738  0.49964549
  0.43569393  0.47212426  0.4321262   0.54538736  0.55400981  0.28780357
  0.60717891  0.44045192  0.44375025  0.41066244  0.46089509  0.56549903
  0.4469832   0.61090301  0.49770068  0.51514707  0.51660249  0.52441062
  0.56544327  0.45403844  0.51349529  0.33231093  0.51709427  0.45153367
  0.55966652  0.52028648  0.41806325  0.38704385  0.45286611  0.55574287
  0.34570508  0.49710599  0.37891076  0.47695505  0.49003471  0.58871104
  0.63826778  0.38180777  0.48029672  0.39925282  0.46560652  0.47425537
  0.50351196  0.45707265  0.57945815  0.65116246]
In [28]:
set_fig()
# ヒストグラムを描く
_ret = plt.hist(uniform_avg10)

中心極限定理

もとのこの分布の真ん中(0.5)を中心とした正規分布が出てくる。

ちなみにこれが正規分布の確率密度関数

$$ f(x, \mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$

イカツイ・・・

基本が大切

平均 $$ \mu = \frac{1}{n} \sum_{i=0}^{n}{x_{i}} $$ 分散 $$ \sigma^2 = \frac{1}{n} \sum_{i=0}^{n}{(x_{i}-\mu)^2} $$ 標準偏差 $$ \sigma = \sqrt{\sigma^2} $$

In [6]:
set_fig()
# 平均0、分散1の正規分布に従う乱数100個
normal_rand = np.random.normal(0, 1, 100)
_ret = plt.hist(normal_rand)
In [29]:
set_fig()
# もっと綺麗な正規分布が描きたいときは、
from scipy import stats

# 便宜的に領域を決めてXをつくる
X = np.arange(-4,4,0.01)
# 値を計算します。
Y = stats.norm.pdf(X,0,1)

# PDF:Probability Density Function 確率密度関数

plt.plot(X,Y)
Out[29]:
[<matplotlib.lines.Line2D at 0x109694828>]
In [8]:
set_fig()

plt.plot(X,Y)
# 全体の面積は1、-2から2までは?
print(stats.norm.cdf(2,0,1)-stats.norm.cdf(-2,0,1))
0.954499736104
In [31]:
set_fig()
# CDF:Cumulative Distribution Function 累積分布関数
Y = stats.norm.cdf(X)
plt.plot(X,Y)
Out[31]:
[<matplotlib.lines.Line2D at 0x1099efc50>]
In [33]:
set_fig()
# boxplotを使ったデータの可視化
_data0 = np.random.normal(0, 1,100)
_data1 = np.random.normal(1, 1.5,100)

data = pd.DataFrame( [_data0, _data1]).T

data.columns = ['mu=0_std=1.0', 'mu=1_std=1.5']
_ret = data.boxplot(return_type='dict', fontsize=20) # 警告抑止のため
In [13]:
set_fig()
seaborn.violinplot(data=data)
Out[13]:
<matplotlib.axes._subplots.AxesSubplot at 0x10a26ad68>
In [34]:
set_fig()
# 平均0と平均2の2つの正規分布から乱数を発生させる
_data0 = np.random.normal(0, 1,100)
_data1 = np.random.normal(5, 1,100)

dual_normal = np.concatenate((_data0, _data1))

_ret = plt.hist(dual_normal,bins=20)
In [16]:
set_fig()
# 分布の詳細な形がわからなくなる
_ret = plt.boxplot(dual_normal)
In [17]:
set_fig()
# ヴァイオリンプロットだと分布の情報が消えない
seaborn.violinplot(dual_normal, orient='v')
Out[17]:
<matplotlib.axes._subplots.AxesSubplot at 0x10a43fc50>

おしまい

またの機会に、検定とかの話もしてみたいと思います